In this paper, we investigate the use of a self-adaptive Pareto evolutionary multi-objective optimization (EMO) approach for evolving the controllers of virtual embodied organisms. The objective of this paper is to demonstrate the trade-off between quality of solutions and computational cost. We show empirically that evolving controllers using the proposed algorithm incurs significantly less computational cost when compared to a self-adaptive weighted sum EMO algorithm, a self-adaptive single-objective evolutionary algorithm (EA) and a hand-tuned Pareto EMO algorithm. The main contribution of the self-adaptive Pareto EMO approach is its ability to produce sufficiently good controllers with different locomotion capabilities in a single run, thereby reducing the evolutionary computational cost and allowing the designer to explore the space of good solutions simultaneously. Our results also show that self-adaptation was found to be highly beneficial in reducing redundancy when compared against the other algorithms. Moreover, it was also shown that genetic diversity was being maintained naturally by virtue of the system's inherent multi-objectivity.
Studying coevolutionary systems in the context of simplified models (i.e., games with pairwise interactions between coevolving solutions modeled as self plays) remains an open challenge since the rich underlying structures associated with pairwise-comparison-based fitness measures are often not taken fully into account. Although cyclic dynamics have been demonstrated in several contexts (such as intransitivity in coevolutionary problems), there is no complete characterization of cycle structures and their effects on coevolutionary search. We develop a new framework to address this issue. At the core of our approach is the directed graph (digraph) representation of coevolutionary problems that fully captures structures in the relations between candidate solutions. Coevolutionary processes are modeled as a specific type of Markov chains-random walks on digraphs. Using this framework, we show that coevolutionary problems admit a qualitative characterization: a coevolutionary problem is either solvable (there is a subset of solutions that dominates the remaining candidate solutions) or not. This has an implication on coevolutionary search. We further develop our framework that provides the means to construct quantitative tools for analysis of coevolutionary processes and demonstrate their applications through case studies. We show that coevolution of solvable problems corresponds to an absorbing Markov chain for which we can compute the expected hitting time of the absorbing class. Otherwise, coevolution will cycle indefinitely and the quantity of interest will be the limiting invariant distribution of the Markov chain. We also provide an index for characterizing complexity in coevolutionary problems and show how they can be generated in a controlled manner.